Highest-Level Formal Solutions

Abstract

In this and later chapters, we are concerned with systems having a singularity of the second kind at some point z ₀ . As in the case of singularities of the first kind, we may without loss of generality assume that z ₀ is any preassigned point in ℂ ∪ ∞. Both for historical reasons as well as notational convenience it is customary to choose z ₀ = ∞ here. Hence we shall be dealing with systems of the form

$$ zx\prime = A\left( z \right)x,{\mathbf{ }}A\left( z \right) = z^r \sum\limits_{n = 0}^\infty {A_n z^{ - n} } ,{\mathbf{ }}\left| z \right| > p, $$

(1)

with Poincaré rank r ≥ 1. Observe that we usually assume the leading term A ₀ to be nonzero, but sometimes we may apply certain transformations, producing a new system with vanishing leading term. In this case we may say that we have lowered the Poincaré rank of the system.